The function \(Q(\mathbf {\alpha _1})\)  is convex [1]} and can be minimized using standard convex optimization algorithms [2]}. Provided that there is adequate overlap, minimization yields the unique finite solution: \(\hat{\mathbf {\alpha }}_{\mathbf {1}}=\textnormal {argmin}[Q(\mathbf {\alpha _1})]\) . Feasible solutions do not exist if all the values observed for a covariate in \(\mathbf {z}\)  are greater or lesser than its corresponding element in \(\mathbf {\theta _z}\)  [3]}.
